L’INSTITUTFOURIER DE ANNALES

N A L E S E D L ’IN IT ST T U

F O U R IE

ANNALES

DE

L’INSTITUT FOURIER

Tanya CHRISTIANSEN & M. S. JOSHI

Scattering on stratified media: the microlocal properties of the scattering matrix and recovering asymptotics of perturbations

Tome 53, n^o2 (2003), p. 565-624.

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565

SCATTERING ON STRATIFIED MEDIA:

THE MICROLOCAL PROPERTIES

OF THE SCATTERING MATRIX

AND RECOVERING ASYMPTOTICS OF PERTURBATIONS

by

T. CHRISTIANSEN and M. S. JOSHI 53, ²

(2003),

^565-624

1.

Introduction.

In this _paper, we

study

^{the structure of the}

scattering

^{matrix on a}

perturbed

stratified medium. In

particular,

^{we show that its main}

part

^{is a} Fourier

integral operator.

^{En route to}

proving

^this

theorem,

^we

develop

^an

improved limiting absorption principle

^{for a}

large

^{class of}

perturbations, using techniques

from Fourier

analysis

and microlocal

analysis.

^{As an}

application

^{of our}

results,

^we prove that the

asymptotics

^{of a}

perturbation

can be recovered from the

scattering

^{matrix at one} energy.

Recall that a stratified medium is a model _{space in} which sound waves

propagate

^{with a} variable sound

speed

^that

depends

^on

only

^one coordinate.

Thus,

^{if we write} the coordinates on R~ as z =

(x, y)

^{with x E and}

y ^E

R,

^{we take the wave}

speed

^to be of the form

co(y)

^and

study

^{the wave}

equation

The first author was partially supported by ^NSF grant ^0088922.

Keywords: Stratified media - Scattering matrix - Inverse problems - Limiting absorption principle.

Math. classification: 35P25 - 81U40 - 35S30.

where We assume that _co is constant for

Iyllarge

^and

that it is

piecewise

^{smooth. Set}

In

general,

^{we do not}

require

^that c+ be

equal

^{to c-.}

However,

^{some of our}

results are

stronger

^when

they

^are

equal.

A

perturbed

stratified medium is a medium in which the variable sound

speed,

c, has the

property

^{that c -} co is well-behaved at

infinity.

The ^case where the

perturbation,

^{c -} co, is

rapidly decaying

^{has been}

studied _{in many}

previous

papers. In

particular, precise asymptotics

^for

(C 2A - (A -z0)~)~/,

^when

f

^{E were} established in

[5].

^{That co and}

the

scattering

^{matrix at fixed non-zero} energy determine an

exponentially decaying perturbation

^was

proved

ⁱⁿ

[20] (co(y) -

^{c± for}

::1:y

&#x3E;

0), [15] (co(y) =

^{c± for}

~y

&#x3E;

yM),

^and

[33] (for

^{co a Pekeris}

profile).

^In

[1], [2]

^a similar result was

proved

^{under more relaxed}

requirements

^on

co. In

particular,

co ^was

required

^to

exponentially approach

^{constants as}

::1:y ---*

^oo, whilst still

requiring

^{that c -} co

exponentially decay.

Here ^we

study

^{the case} where the

perturbation,

^{c -} co, has an

asymptotic expansion

ⁱⁿ

homogeneous

^{terms at}

infinity.

Under certain conditions on c and _co made ^more

precise

^{in Section}

2,

^we show that the

scattering

^{matrix for}

^c20

^{is a Fourier}

integral operator

and describe its

singular

^set.

Moreover,

^{we show that the}

asymptotics

^{of the}

perturbation

can be recovered from the

scattering

^{matrix at fixed} energy. We also establish the

leading

^{term of the}

asymptotics

^{for the}

limiting absorption principle.

Our results use

techniques developed by

Joshi and Sa

Barreto, [21], [22], [23], [24], [25],

^to

study

^inverse

problems

^{in other}

settings.

^These

build ^on work

by

^Melrose

[26],

^and Melrose-Zworski

[27],

^{on the structure}

of the

scattering

^{matrix on}

asymptotically

^Euclidean spaces. As in those inverse

results,

the fundamental idea here is to

compute

^the

symbol

^{of the}

scattering

^matrix

by solving transport equations along geodesics

^{on the}

sphere

^at

infinity.

^These

equations

express the

propagation

^of

growth

^at

infinity.

The

analysis

^here

is, however, considerably

^{more involved as the}

unperturbed

^wave

speed,

co, is not smooth on the

compactified

space obtained

by adding

^the

sphere

^at

infinity,

^{even when}

co(y)

^{is a smooth}

function of _y. This is because _co does not have nice

asymptotics in Izl.

Therefore _co is well-behaved on the

compactified

space

only

^{after the} space has been

blown-up

^{on the}

equator

^at

infinity.

This manifests itself in our

analysis by requiring

^the

geodesic

^{flow at}

infinity

^to be refracted and reflected

by

^the

equator.

^{It was also seen in}

[5]

^{that it makes the}

asymptotics

in the

limiting absorption principle

^{much more}

complicated.

^{There is a}

certain

similarity

^{here with}

many-body scattering,

compare, e.g.

[29].

^There

the

scattering problem

^is

complicated by

^the presence of a

potential

^that

does not

decay

in certain directions and thus _appears ^{as a}

spike

^{on the}

sphere

^at

infinity.

^{This causes} refractions and reflections of the

geodesic

flows

[29]. Indeed,

^{the case where} c+ ^{= c-} bears much resemblance to the

many-body

^case.

However,

^when

c+ ~

^{c_ , there are}

effectively

^different

energy levels in the two

hemispheres,

and this introduces new

complications

which are not

present

^{in the}

many-body setting,

and much of this _{paper is} dedicated to

coping

with these

complications.

In Section 5 we define the

scattering matrix,

^{and its "main}

part."

When the

operator

⁺

",,2)

^{has no}

eigenvalues

^{as an}

operator

^on

co 2dy)

for all real _x, then the main

part

^{of the}

scattering

^matrix

is the same as the

scattering

^matrix.

Below we refer to the

hypotheses (HI)

^and

(H2).

^{For full}

details,

see Section

2,

^but

roughly speaking hypothesis (HI)

^{is that c+ = c-, c}

and _co are

smooth, and c - co) x C(I + z ~ ) -2 . Hypothesis (H2)

^{is that}

Our first main result is

THEOREM 1.1.

Suppose

c, co

satisfy

^the

general assumptions

^of

Section

2,

and either

hypothesis (Hi)

^or

(H2). ^Then,

^if c+ = c_, the main

part

^{of the}

scattering

^{matrix is a} zeroth order Fourier

integral operator

associated with broken

geodesic

^{flow at time 7r. If c-} &#x3E; c+, then the main

part

^{of the}

scattering

^{matrix is a sum} of Fourier

integral operators

associated with the

mapping

and the

mapping

and

Here,

^when c+ - ^c-, the

geodesic

^{flow is broken at the}

equator

(w, 0)

^{C This can be}

^compared

^{to the situation for the}

Laplacian [27],

or a

perturbation

^{of the}

Laplacian

^{to an}

integral

power

[6],

^{on a manifold}

with

asymptotically

^Euclidean

ends,

^{where the}

scattering

^{matrix is a zeroth}

order Fourier

integral operator

associated ^to

geodesic

^{flow at time 7r on the}

the

boundary

^"at

infinity."

An additional

analogy

^{is to}

3-body scattering,

where the three-cluster to three-cluster

part

^{of the}

scattering

^{matrix is a}

sum of Fourier

integral operators

associated to broken

geodesic

^{flow at time}

7r

[29].

Other results on the structure of the

scattering

^{matrix in}

n-body scattering

may be found in

[30].

Further results on the structure of the

scattering

^{matrix are}

given

ⁱⁿ

Proposition

^5.2.

Our central inverse result is

THEOREM 1.2. -

Suppose

^{c and} co

satisfy

^the

general assumptions

of Section

2,

^{as well as either}

hypothesis (Hl)

^or

(H2),

and n &#x3E; 3.

Then,

if c+ - c_, the

asymptotic expansion

^at

infinity

^{of c - co is}

uniquely

determined

by

co and the transmitted

singularities

^{of the main}

part

^{of the}

scattering

^{matrix at fixed non-zero} energy.

If c+

^{c-, then the}

asymptotic expansion

^is

uniquely

determined

by

co and the reflected

singularities

^of

the main

part

^{of the}

scattering

^{matrix at fixed non-zero} energy.

The reflected

singularities

^are those associated to the

mapping (-W, Wn), ^and,

^{for c+ - c-} the transmitted

singularities

^are

those associated to the

mapping

^-We

Corollary

8.1 shows that knowl-

edge

^of c+,

c-, A

^{and the}

singularities

^{of the}

(absolute) scattering

^matrix

for _any fixed non-zero energy determine the

asymptotics

^{of c.}

Following

^the

approach

^to

studying

^the

scattering

matrix introduced in

[27],

^{in Section 7 we construct a}

parametrix

for the Poisson

operator.

This is a

key part

^{of our}

proofs,

^{as it} facilitates an

understanding

^{of the}

singularities

^{of the}

scattering

matrix. We work

particularly by adapting

^the

techniques

^of

[22]

^{which are}

essentially

^a concretization of the

approach

introduced in

[27]. ^However,

the different behaviour of the

unperturbed operator

^{in different}

regions

^at

infinity

^{means that the}

analysis

^is

considerably

^{more involved.}

To _pass from a

parametrix

^to the actual Poisson

operator,

^{we need a}

good understanding

of the behaviour of

(A - (A - iO)2c-2)-1 f

^at

^infinity,

when

f

^E

(z)-OO L2(IRn)

^and

(1 - cp(y))f

^{E for}

^{some 0}

^E

In

practice,

^{we shall}

apply

^this

understanding

^{to the error}

arising

^from

the

parametrix

of the Poisson

operator.

^When c+ - c-, ^{we can} obtain the necessary results

by modifying

^some

n-body

results of

[14]

^and

[28].

However,

^when c+ ^c- these results no

longer apply,

^{and we}

develop

new

techniques.

The essential idea of these

techniques

^{is to}

repeatedly develop

^better

approximations

^with

improving

smoothness

properties.

^Thus

Sections 9 and 10 are devoted to

understanding (A - (A -

allowing

^{us to} finish the

proof

of Theorem 1.1. In

particular,

^we prove the

following limiting absorption principle:

THEOREM 1.3. ^- Let c and co

satisfy

^the

hypotheses

^{of Section 2 and}

hypothesis (HI)

^or

(H2).

^For any X ^E

C°° (~~ -1 ),

^and

f

^E

such that

(1 - 0(y))f

^{E for}

^{some 0}

^{E ure have}

where ao e and ui E for all e &#x3E; 0.

Here

C°° (~~ -1 )

^{is the space} of smooth functions

vanishing

^{in a}

neighbourhood

^{of the}

equator

^{and in a}

neighbourhood

^E

An announcement of some of these results and an outline of

part

^of the

proof

^{can be found in} the lecture notes

[7].

A note on

organization:

We need Theorem 1.3 _{to prove} Theorems 1.1 and 1.2.

However,

^{since the}

proof

of Theorem 1.3 is rather involved and

uses different

techniques

from much of the remainder of the _paper, we defer its

proof

^to Section 10. Sections 2 and 3 contain

preliminary information, stating assumptions, fixing notation,

^and

recalling

^some results of other papers. In Section 4 we define the Poisson

operator,

^{which we use in} Section 5 to define the

(absolute) scattering

^{matrix. Also in Section 4}

we prove the existence of the Poisson

operator

^and prove Theorem

1.1, though

^{we use results}

proved

^{later in the} paper. The

proof

that the Poisson

operator (and

^{thus the}

scattering matrix)

^{is well} defined is deferred to Section 6. In Section 7 ^we construct

P,

^an

approximation

of the Poisson

operator P, proving Proposition

^{4.2. Theorem 1.2 is}

proved

in Section

8, using

the construction of

P

and Theorem 1.3.

Finally,

Sections 9 and 10 contain results about

C-2(À - iE)2)-1, culminating

^{in the}

proof

of Theorem 1.3.

We are

grateful

^{to Fritz}

Gesztesy

^for

helpful

conversations and

providing

useful references and ^to Jim Ralston for

helpful

discussions. We thank the London Mathematical

Society

^for

supporting

this collaborative

research

through

^{its small}

grants

scheme. We also thank the referee for his

or her careful

reading

^{of an earlier version. The}

resulting helpful

^comments

have

improved

^the

exposition

^{of this} paper.

2.

Assumptions

and notation.

Throughout, z =

^{E x R.}

Both sound

speeds

^{c and} co

satisfy

^{0 cm x c, cM oo.}

Moreover, co(y)

^is

piecewise

smooth and there exists a finite _yM so that

co(y) =

^{c± when}

::1:y

&#x3E; yM, with c- &#x3E; _c+. In

addition,

all derivatives of co ^are bounded

except

^at

finitely

many values of _y. This includes the ^case where co is

piecewise

^constant.

We

require that,

away from the

hypersurface 01,

^{c - co be}

smooth outside of ^a

compact

^set

K,

^{and for}

simplicity

^{we choose} yM ^so that K C

R n-1

^x

[-yM, yM]. ^Moreover,

^{we make}

requirements

^{on the}

behaviour of c - co at

infinity.

^We

have,

^for

Y =1= 0,

for _any N and _any multi-index _a,

where qj

^E

0) ~ ) .

^{Here we}

use the notation that is the _space of smooth functions on X that have all derivatives bounded. We shall take J to be at least 2

everywhere, although

^{sometimes we shall}

require

^{it to be}

larger.

^{Some of our results}

hold under less restrictive

hypotheses.

Additionally,

^we shall often use one of the

following hypotheses:

(HI)

^J

= 2,

c+ ⁼ c-, ^c and _co are smooth.

(H2) J &#x3E;

^4.

We ^warn the reader that the choice of the total _space dimension to be

rt rather than n+ 1 ^{is in}

disagreement

^with

[5]

^{and many other papers on the}

subject.

^{We use} the notation

(w ~

^_

(1 + IwI2)1/2. Throughout,

^c shall stand for a small

positive quantity

^{and C for a}

positive constant,

either of which may

change

^{from line to} line. For w E

R’, b(w, t)

^-

means

for _any N and t E

K,

^{K a}

compact

^{set in some manifold.}

3.

Spectral theory

^of

co0.

In order to define the

(absolute) scattering

matrix for we will need some

understanding

^{of the}

generalized eigenfunctions

^of

co0

^and

^{c A ,}

and in

particular

^{of the} space that

parameterizes

them. Further details ^can be found

in,

^for

example, [3], [13], [32], [34].

The

operators co0

^and

^c20

^are

formally self-adjoint

^on

L (R , C02 dz)

and

c-2dz), respectively,

^{and have}

unique self-adjoint

extensions.

Roughly speaking,

^the

spectral

^{measure of can be}

given

ⁱⁿ

terms of two kinds of families of functions. At fixed _energy

A,

^{the first}

is

parameterized by ~~ -1.

^Here

(Compare [32,

^Section

2.1~;

^{we are}

modifying

somewhat the notation of

[32].)

^The

generalized eigenfunctions

^are

À, w),

^{where for} &#x3E;

0,

and

0±

^satisfies

Moreover,

^as y - oo,

and as y - -cxJ,

where when

1 /c2 -

^{+ 0 we take the} square root ^so that the

right

hand side of

(8)

^is

exponentially decreasing.

The function

ø-

^is

similarly

determined: _{as y - -oo,}

and ^2013~ oo,

Properly normalized,

^these

generalized eigenfunctions

^{appear in the} spec- tral

decomposition

^of

A second

type

^of

generalized eigenfunction

^{comes from}

eigenvalues

^of

CÕ(11:2 + D 2)

^on

^{L2 (R, C-2 dy),}

^{if there are any. If there are any}

eigenvalues,

let

A~(~) A 2 (K)

^...

~~(,~) (~) C2K2denote

^the

eigenvalues

^of

cÕ(11:2 + D)).

^{There are}

only finitely

many

(perhaps no) eigenvalues

^for

fixed K and the number is

nondecreasing

ⁱⁿ

^~2. Additionally,

^{if K} &#x3E; 0 and

Aj

&#x3E;

0,

^then

~

&#x3E;

0;

^{this can be seen via an}

integration by parts argument

(see,

^e.g.,

[5,

^Sect.

2.2]).

Figure

^{-L The}

spectrum

⁺

Dy ) ,

^{for c+ c_ ,}

^{min co}

^c+.

Let KJ

be the smallest

positive

number such that

CÕ(K2

^-I-

D~)

^has

j eigenvalues

for all K &#x3E;

KJ. Let Kj

be the inverse function of

Àj (with

the same

sign),

^and

let tj

^-

() - c+(,o)2.

^{The are called}

thresholds of Let

T(A)

be the number of

thresholds tj

^{less than}

^A2.

For 0

where E satisfies

and note that

(co0 -

0. The functions ^are

exponentially decreasing,

^{as can be seen}

by noting

^{that for} &#x3E; YM,

they

^are

^L~

^solutions

At _energy level

~2,

^{we can}

parameterize

^the

generalized eigenfunctions

of

co0 by Sn- 1

^and

T(A) copies

^of

S,-2 .

The continuous

spectrum

of

c2 A

is

parameterized by

^{the same} space ^as that of

C2 A. ^Therefore,

the

(absolute) scattering

matrices of and are

operators

^from

L 2(Sn-1) L2 (Sn-2)

into itself. In

[5]

^a definition of the

scattering

matrix is

given

^{in terms of the}

generalized eigenfunctions. ^Here, however,

it will be ^more useful to define the

(absolute) scattering

^matrix

using

^the

Poisson

operator,

^{which we shall do in Section 5.}

4. The

Poisson operator.

The Poisson

operator

^{is defined as an}

operator

(Again,

^{we have}

C°° (~~ -1 )

^because

S’-1 parameterizes part

^{of the contin-}

uous

spectrum.)

In order to define the Poisson

operator,

^we introduce the notion of ^an

"outgoing"

^{function in this}

setting.

DEFINITION 4.1. A function u E

(Z)1/2+EL 2(R n)

will be called

outgoing

^{if it has a}

decomposition u

⁼ Uo + + U with the

following properties:

for _any c &#x3E; 0.

PROPOSITION 4.1. -

then for A E

R, ~ ~ 0,

^{there is a}

unique u

^{such that}

and,

^at

infinity,

where v is

outgoing

^{as defined in} Definition 4.1 and the

fj

and Kj ^{are as} defined in Section 3.

We will

give

^the

proof

^{of the existence of such a u in Section 4.1. We}

postpone

^the

proof

^{of the}

uniqueness

^{to Section 6. This}

proposition

^allows

us to define the Poisson

operator, P(A).

DEFINITION 4.2.

then

P(A)g =

^{u, where u is the u of}

^Proposi-

Definition

5.1, using Proposition 5.1,

^{defines the}

(absolute) scattering

matrix via the Poisson

operator.

4.1. Existence of the Poisson

operator.

Here we prove the existence

part

^of

Proposition 4.1, using

^{some results}

that ^we _prove later in the _paper. The first

step

^is the construction of an

"approximation"

of the Poisson

operator,

^{which is carried out in Section 7.}

Other, simpler, proofs

^{of the existence}

^part

^of

Proposition

^{4.1 are}

available,

but this one facilitates the

proofs

of Theorem 1.1 and 1.2.

For w E

sn-l,

^let

8w (0)

be the distribution such that

and

similarly

^for

PROPOSITION 4.2. - There is a distribution i such

that, for ,

where

X(y)

^E

C~(Jae)

^{is 1 for yM + 1 and a,}

^{3

^{are any} multi-indices.

Moreover, distributionally

^as

lzl

^{- oo,}

where

ho (w, 0)

^E _c ^X

Sn- 1).

^{If c+ - c-,}

^ho

^is the Schwartz kernel

c

of a Fourier

integral operator

associated to broken

geodesicflow

^{at time 7r.}

If c- &#x3E; c+, then

ho is

the Schwartz kernel of the ^sum of Fourier

integral operators

associated with the

mapping

and the

mapping

and

For j

&#x3E;

0, distributionally as Izl

2013&#x3E; oo,

Here

hi

^E

v,(sn-2

^X

Sn-2)

^is the Schwartz kernel of a Fourier

integral operator

associated with the

antipodal

map ^on

Sn-2.

We will _prove this

proposition

^{in Section 7.}

We also use

PROPOSITION 4.3.

If f

^E

(z~ 3~2+EL2(I~n)

^{for every E} &#x3E; 0 and

J &#x3E; 2,

^{then u =}

(A - (A - iO)2C-2)-1 f

^is

outgoing

^{in the sense of Defini-}

tion 4.1.

This

proposition

^{will be}

proved

^{in Section 9.}

Proof of the Existence Part of

Proposition

^{4.1. Let}

and let For

Then

and u has an

expansion

^at

infinity

^as

required

ⁱⁿ

Proposition

^{4.1. Conse-}

quently, Q

⁼

P(a),

the Poisson

operator.

^D

5. The

scattering

matrix and the

proof

^of

^Theorem

^1.1.

We will define the

(absolute) scattering

^matrix via the Poisson

operator.

^In

doing

so, ^{we assume} that the Poisson

operator

^is

uniquely determined;

^{we will} prove this in Section 6.

We shall use the

following lemma,

which will be

proved

^{in Section 9.}

LEMMA 5.1. ^-

If f

^E

(z~ -°° L2 (II~n ),

^then

for y

^E

K,

^{K c R}

compact,

where bj

^{C and vi E}

x K) for any E

&#x3E; 0.

Moreover,

for

1 j - T(A)

^the

uj in

the definition of

outgoing

^{functions can be taken}

To define the

scattering operator,

^{we need} PROPOSITION 5.1. Let

for K a

compact

^{set in}

~~ -1. ^Then,

^there

exists j

such that for 9

where

uK

^{I for} any ^c &#x3E; 0.

Let _y E

Ky, Ky

^{C R}

^compact,

and let 9 =

Then,

^{there exists} I ^{such that}

as Ixl -~

^oo,

where ¹

Proof. We use the ^same notation to stand for an

operator

^{and its} Schwartz kernel. Since

the first

part

^{of this}

proposition

follows from

Propositions

4.2 and 4.3 and Theorem 1.3. The second

part

^{of the}

proposition

^follows

again

^{from the}

identity (11), Propositions

^{4.2 and}

4.3,

and Lemma 5.1. 0

This information about the Poisson

operator

^{allows us to define the}

(absolute) scattering

^matrix

A(A).

DEFINITION - 5.1. ’71t B. ^- B The

(absolute)

~

scattering

^matrix

A(A)

^is -

^given,

’71t B. B

where for _any

compact

^{set K C}

~~ -1,

^{is as in}

Proposition 5. l, and,

^{for as in}

Proposition

^{5. l.}

We remark that this definition differs

slightly

^from the absolute scat-

tering

matrix discussed in

[5]. ^However,

^{as the two differ}

by

^a

straight-

forward

normalization,

^{we shall use} this definition here both to

emphasize

the similarities with the absolute

scattering

^{matrix as} defined in

[26]

^and

because it is more convenient for the inverse results.

For

completeness,

ⁱⁿ

Appendix

^{A we outline a}

proof

^that

A(A)

^can

be extended to a bounded

operator

^on

L 2(S,-I) EBf(À) j,2 (~n-2 ) _

For fixed

A, A(A)

^{is a matrix}

(Aij(À)), ^{0 - i, j} T(A),

^{with the}

Aij

operators.

We shall call

Aoo (a)

the "main

part"

^{of the}

scattering

^matrix.

If the

operator c5(Dy

⁺

~2)

^{has no}

eigenvalues

^on

c¡;2dy),

^{the main}

part

^{of the}

scattering

^matrix is the entire

scattering

^matrix.

We ^{can now} _prove Theorem

1.1,

^{on the structure of the}

(absolute) scattering

^matrix.

Proof of Theorem 1.1.

Again,

^{we use the}

identity (11). Using

^the

definition of the

scattering

^{matrix and Theorem}

1.3,

any

singularities

^{in the}

main

part

^{of the}

scattering

matrix must come from

Po.

^Then

Proposition

4.2

gives

^{the structure of the}

singularities

^{of the}

scattering

^{matrix. D}

Finally,

^we conclude this section with a

proposition

that describes the

singularities

of the other entries in the

scattering

^matrix.

PROPOSITION 5.2. ^- Let _{c, co}

satisfy

^the

general

conditions of Section 2 and either

hypothesis (Hl)

^or

(H2).

^Let

A(A) = (Aij(À)), ^{0 , i, j -} T(A).

Then, for j

&#x3E;

0,

^{is a Fourier}

integral operator

associated with the

antipodal mapping

^on

sn-2,

^{and for i not}

equal

^to

j, Aij (A)

^{is a}

smoothing operator.

Proof.

Using

^the

identity (11),

^Lemma

5.1,

and Theorem

1.3,

^the

singularities

^{of the}

scattering

matrix arise from Then

Proposition

^4.2

and the definition of the

scattering

matrix show that the

singularities

^of

the

scattering

^{matrix are as claimed. D}

6. Proof of

Proposition

^{4.1 :}

uniqueness.

In

proving Proposition 6.1,

^{we shall use some} results of Weder

[31], [32] (See

^also

[11]).

^{We recall some} of his results below. Let A -

(-i/4)(z ^Vz

⁺

z).

We define the commutator

[A - ,B2/c2,A]

^{as a}

quadratic

^form

(See

^the

proof

of Theorem 5.4 of

[32]). By [31,

^Lemma

3.1]

for all A &#x3E;

0, p

&#x3E;

-~2/c2 ,

^{there is a}

compact operator K,

^a

compact

interval A

containing

p, and

(3

&#x3E; 0 such that

where

En = A 2C-2)

^{is the}

spectral projector

^{for 0 -}

^A2c-2.

The

following proposition

^{and its}

proof,

included for the convenience of the

reader,

^are

essentially adapted

^from

[2,

^Lemma

4.17].

PROPOSITION 6.1. for _every c &#x3E; 0 and

(A - A 2/C-2) U

⁼

0,

^{then u - 0.}

Proof.

By

the results of

[31], [32],

^{there is no} nontrivial

L2 null

space of A -

~2 /c2,

^{so it suffices to} show that u E

and let 4) E be 1 in a

neighbourhood

of 0. Note that

where

Then

Since

we have

Here and below

CD

^{is a constant that} may

change

^{from line to line which}

depends

^on

D,

and also on A and c, but which is

independent

^{of 6 and E.}

Since, using (13),

^a similar bound holds for we

obtain from

(14)

Since ⁼

0,

^{we have}

The bound can be _seen, for

example, by using

^the

Helffer-Sjostrand representation

^of

However, using (12),

the fact that 0 is not an

eigenvalue

^of

L,

^and

[9,

Lemma

4.2],

^{we obtain}

for some

{3l

&#x3E;

0,

^{if the}

support

^{of (D is chosen}

sufficiently

small. Thus we

have, using (15)

^and

(16),

^and

choosing E sufficiently small,

Therefore

Using (16),

this shows that for

sufficiently

^{small E} &#x3E;

0, IluEó II

^{is bounded}

independent

^of

6,

and thus u E

L~(R~),

^{and u - 0. D}

This allows ^us to show the

uniqueness

^of

"outgoing"

^solutions.

PROPOSITION 6.2. ^- Given

f

^{E there is at most one}

outgoing

- 1- - - -

Proof. ^-

Suppose

^{there are two such u. Then}

by considering

^the

difference ^{we can} reduce to the case where

f -

^{0. Then}

where E for all E &#x3E; 0.

Using

the facts that

f fj (y) f ~ (y)dy

^{= 0}

if j 54

^{k and}

fj

^is

exponentially decreasing,

^this

implies

that as R - ^o0

Since uj ^E

(y) -°° (x) 1~2+EL2 (Ilgn)

^{and uo E}

(y~ 1~2+E (z~EL2 (II~n),

^{we have}

uouj

^E

(y~-°° (z) 1~2+ZEL1 (I~n). Therefore,

^the

right

^{hand side of}

(17),

considered as a function of l~ for

large R,

^{is in}

R’/2+2"L’(R+),

^{so that}

Now _suppose ^we know that _uo,

Uj E

^{for some}

13

&#x3E; 0.

Then, using (17) again,

^{since and uoeo,} ujej,

e f

^E

(z)/3+t Ll (IRn),

^{we obtain} uo, Uj ^E

(z~~~2+EL2 (Il~’~)

^{for any c} &#x3E; 0.

By repeating

^this

argument,

^we obtain that _uo, Uj ^E

(z)8 L2(IRn)

^{for any}

6 &#x3E;

0, 1 j T(A).

^{Thus u E}

(z)6L2(R")

^for

^{any 6}

&#x3E; 0.

By

^the

previous

proposition, u -

^{0. D}

Proposition

^6.2

gives

^the

uniqueness part

^of

Proposition

^4.1.

7.

The approximate ^Poisson operator P.

In this section we prove

Proposition

^{4.2: we construct an}

approxima-

tion of the Poisson

operator.

^{We will use the}

following

^lemma.

LEMMA 7. 1. - For

f

^{E the}

boundary

^value

problem

with

boundary

^conditions

has a

unique

solution in

Proof. This

boundary

^value

problem

^can be reduced to the form

This has a solution if the null _space of the

adjoint operator

^is

trivial,

^and

the solution is

unique

^{if the}

only

solution of the

homogeneous equation

^is

the trivial one.

The

adjoint operator

^{is the}

operator

with domain

and

Suppose g

^{is a} nontrivial element of the null _space of the

adjoint operator.

Then

Using

^the

boundary conditions,

^we find that this is

Thus

A similar calculation shows that the

original operator

^{has no nontriv-}

ial null _space. ⁰

In our

proof

^of

Proposition 4.2,

^{the existence of an}

approximation

^to the Poisson

operator,

^{we shall} concentrate on the construction of

Po.

^This

is the most

complicated component

and also the ^one of

primary interest,

since our inverse results involve the main

part

^{of the}

scattering matrix,

The construction ^{owes a}

great

^{deal to that of}

[22],

^{and we refer the}

reader ^to it for full details.

We will show how to construct

Po (z, A, w) when Wn

&#x3E;

0;

^{the case of}

wn 0 is

quite

similar. The construction involves

solving

away ^errors at

infinity.

Since the model

operator

has different behaviour

depending

on the

region

^"at

infinity" ( y

&#x3E;

H

yM , or y

-ym)

^the

techniques

^involved

necessarily depend

^{on the}

region

ⁱⁿ which z lies. The

proof

^is

additionally

^{divided into two subcases:} &#x3E;

0,

^and

1 /c2 -

^0.

Roughly speaking,

the second subcase

corresponds

^to

total internal reflection and for these values

Po (z, A, 1J)

^is

exponentially decreasing in y

^as Y ---* -00. In this subcase we can handle the entire

region

y yM at ^once. The first subcase

corresponds

^to

angles

of incidence in which some of the wave is

transmitted,

and for these values of 1J,

Po

^is

oscillatory

^at

infinity

ⁱⁿ all directions.

Finally,

^{there is a third}

division, involving

the construction of

Po

^near

the

points

^{where it is}

singular

^at

infinity (and

^{where the}

singularities

^{of the}

scattering

^matrix

arise).

^{This we defer to} Section 7.1.

In an

attempt

^{to make the}

proof

^more

readable,

^{we outline our}

construction of

1. y

&#x3E; yM, the

"upper hemisphere"

a)

"incident"

b) "reflected,"

away from the

singular point = ( -~, wn)

a) y

^-yM, the "lower

hemisphere,"

away from the

singular point

( "transmitted" )

Section 7.1. Construction near

singular points.

The numbers

correspond

^to

numbering

^{of the}

paragraphs.

1. Let c.v ⁼

(w,

^{with Wn} &#x3E; 0. In our construction of the

approxi-

mation to the Poisson

operator Po,

^we

begin

with the function

A, w)

which is defined

by (5)-(8).

^Note

that,

up to ^a constant

multiple

^which

depends only

^on n,

A,

^and c~ ,

(Do

^is the Schwartz kernel of the

(partial)

Poisson

operator, Po,,O,

^{for when cv is in the upper}

hemisphere

^of

^Sn-1.

We use this as our

starting point, adding

^or

subtracting

^{terms to cancel}

the ^errors that result when we

apply ^{c20 - A 2}

^to

4bo.

When _y &#x3E; we use the

techniques

^of

[22]

^{to construct}

^P.

^Note

that when we

apply ^{c20 - A 2}

^{to we obtain an error}

which, for y

&#x3E; yM, is of the form

where E

S;h2g.

^{Here we say that b E if}

b -

with bk

^{smooth in} w, and call it a

polyhomogeneous symbol

^of

order - j .

We think of as an "incident" error and as the "reflected" error.

Ia. Note that if

b(z, A,c~)

^{E then for} y &#x3E; YM,

Of the terms in

parentheses

^{on the}

right,

^{the first is of the}

highest order,

E The others are in

,S’~h9 2 .

Suppose

^our construction to some

point

has resulted in an error of the form

similar to that of

(20).

^{To remove the term}

from the _error, we look for with

,

Using (21),

^{if we} then subtract

from ^{our current}

approximation,

^{the error term is}

again

of the form

(22),

but the coefficient of

e’)"*’/’+

in the new error will vanish to one order faster at

infinity.

^{We choose}

bI,_j+l ( ~ z ~ , ^{À, c~)}

^so that it is smooth at

zllzl I

^{= w}

in order to

keep

^the

right

coefficient of

e’Alzl/c+

ⁱⁿ the distributional

asymptotic expansion. (Here

the "I" in the

subscript

stands for "incident."

Later we shall see "R" for "reflected" and "T" for

"transmitted.")

To solve

(23),

introduce the

"polar"

coordinates

(s, 0),

^{centered at w,}

in

place

^{of That}

is,

let s be the

geodesic

^{distance on the}

sphere

^from

w to

z/lzl

^and

let 9

be

angular

coordinates about w. Then

equation (23)

can be

solved, just

^{as in}

[22,

^Section

2], giving

We are

abusing

^notation

here, using

^{the same notation for}

A, w)

^and

bI,_~+1 (s, B, ~, c.~),

^and

similarly

^for

dI,-j.

^{Note that as}

long

^as

z/lzl

^{is in the upper}

hemisphere

^{we are} away from -1J so the

transport equation

^{has a} smooth solution.

Moreover,

^since

-(Do(z,A,w)

^{is smooth in}

We find

iteratively

^{and then use} Borel’s lemma to

asymptotically sum obtaining

^a

bj

^{such that}

when _y &#x3E; yM . Note that the construction of

bI

^{has not}

changed

^a2.

Ib. We will

apply

almost the same

technique

^{to solve away the error}

yM, away from

zllzl == (-w, wn).

^{Here we will}

use solutions to the

transport equation

^{where we choose} the initial condition at

y/lzl = 0,

^{and the}

solutions,

ⁱⁿ

analogy

^to

(24),

^are of the form

Here s is the distance on from 0 ⁼ to the

point

^and

6

is the

angular

coordinate about

(ZJ, 2013~~).

^{The value s =} sRo

corresponds

^to

On

⁼

0,

^and

CR,j depends only on w and B

and will be determined below.

We

postpone

^to Section 7.1 discussion of the form of the

parametrix

^near

= (-LV,

^the

singular point.

IIa. In the lower

hemisphere,

^{we use a similar}

technique

^if

1/c2 -

&#x3E; 0. Here the error term is of the form

where _{aT E}

Again

^we have solutions like

(24)

^{to the}

^transport

equation, although

^{this time s measures} the distance on the

sphere

^from

the

point

^{We will have an} additional term away

from of the form

where is a solution ^{to a}

transport equation

^like

(23):

Again, s

^is the distance on

S’-’

from and sT,,

corresponds

^to the distance at ⁼ 0.

Y

The constants

(in s) CR,j

^and

CT,j

^{are to} be determined. Of course

their values affect

subsequent

^{errors and thus}

subsequent

IIb. We will use a different

technique

to construct the solutions when

Iyl

yM . We

point

^{out that if co is not}

smooth,

^for

example,

^{if it is}

piecewise constant,

^{we should}

expect

^{it to be}

impossible

^{to find a smooth}

Poisson

operator

^{on R~. We choose our}

approximations

^{so that}

Po

^is

Cl

on R’.

The values of

CR,j,

^and

CT,j

^are determined

by

^{solutions to}

boundary

value

problems

^{that arise in}

constructing

^the

parametrix when jyj

as described below.

When jyj

^the errors are of the form

We look for an

approximate

solution of the form

The is of lower order and is included to

improve

^the

regularity at y = ~ yM .

^{We will} suppress the

dependence

^of

bR,-j ,

^and

bM,-j

^{on A and w to}

^simplify

^{notation. Note that}

Therefore, for y ~

^{yM , to solve away an error} of the form

we look for

bM,-j

^{such that}

The

boundary

conditions which must

satisfy

^{come from}

matching

with the solutions in the

top

and bottom

hemispheres

^{in order to}

get

^a function which is

C 1

"at

infinity". They

^are

where is known

(it

is determined

by integrals

^over

portions

^of

geodesics

^{of and}

CRj

^{are to be} determined and ^are

independent of y,

^{and so can be treated as} constants in

solving

^the

boundary

value

problem. They

^can be eliminated from this set of

equations, resulting

in a

boundary

^value

problem

^{of the}

type

considered in Lemma

7.1,

^which

guarantees

^{us a}

unique

^{solution to the}

problem

^when

1 / c2

This then determines

bR,-3

^and

bT,-j,

^since

CT,j

^{and are determined}

by bM,-j.

We remark that if

0,

^{then =}

this will be

important

^when

studying

^the

inverse

problem.

In order to ensure that our function will be

C’

^at _{y - yM} and at y = -YM ^we will add an additional term whose total contribution will be of order

Let X

^E

c~(Iae), X(t)

^{= 1}

for It I

^{1 and}

X(t)

^{= 0 for}

I t &#x3E;

^{2. Let}

Note that

by

^{our choice of}

b3,

’YU3, ^and qLj ^{all have}

leading

^order

Now,

^let

For jyj

^{yM ,} this determines the

approximate

solution of the form

(26).

III. If

0,

^{then we use a}

slightly

different method for

finding

^the

approximate

^solution

when y x

yM .

Here,

^{in a manner similar}

to that used

for lyl

yM

above,

^{we solve} away the error term

by using

^an

approximation

of the form

where